Finite Group Databasebeta

← #30650713

Subgroups — #30650713

Order 16256.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550576C211yesmaximal, characteristic, complemented
2#22550529C211yesmaximal, characteristic, complemented
3#22550244C211yesmaximal, characteristic, complemented
4#22550216C211yesmaximal, characteristic, complemented
5#22550205C211yesmaximal, characteristic, complemented
6#22550177C211yesmaximal, characteristic, complemented
7#22550119C211yesmaximal, characteristic, complemented
8#22550102C211yesmaximal, characteristic, complemented
9#22550020C211yesmaximal, characteristic, complemented
10#22550020C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549946C211yesmaximal, characteristic, complemented
13#22549876C211yesmaximal, characteristic, complemented
14#22549871C211yesmaximal, characteristic, complemented
15#461×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861202—21no—
22#16861201C2211yescharacteristic
23#16861194—21no—
24#16861194—21no—
25#16861192—21no—
26#16861185C2211yescharacteristic
27#16861185C2211yescharacteristic
28#16861184—21no—
29#16861182—21no—
30#16861180C2211yescharacteristic
31#16861180C2211yescharacteristic
32C4×D508C2211yescharacteristic
33C4×D508C2211yescharacteristic
34C4×C22⋊D127—21no—
35#16861167C2211yescharacteristic
36C4⋊C4×D127—21no—
37C4⋊C4×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861133C2211yescharacteristic
42#16861133C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C4⋊Q8×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—22no—
61#14523249—41no—
62#14523249—41no—
63C2×D508—21no—
64C2×D508—21no—
65C2×D508D411yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74C2×C4×D127C2311yescharacteristic
75#14523233—22no—
76#14523231—41no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C8×D127—41no—
80C4⋊Dic127D411yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96(C2×D127)⋊C4D411yescharacteristic
97C4×Dic127—21no—
98C2×Q8×C127C2311yescharacteristic
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127—21no—
102SD16×C127—41no—
103SD16×C127—41no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127—21no—
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127C2×D411yescharacteristic
113C22×D127C2×D411yescharacteristic
114D508—21no—
115D508—41no—
116D508—42no—
117D508—42no—
118D508#4112yes—
119C22⋊D127—42no—
120C22⋊D127—81no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C4×D127—21no—
127C4×D127—21no—
128C4×D127—21no—
129C4×D127—21no—
130C4×D127—42no—
131C4×D127—42no—
132C4×D127—42no—
133C4×D127—81no—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127—21no—
137Q8×C127—21no—
138Q8×C127—21no—
139Q8×C127—21no—
140Q8×C127—21no—
141D4×C127—21no—
142D4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—81no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—22no—
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—21no—
162C2×D127—21no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127C2×#4111yescharacteristic
171C4×C127H(5,𝔽2)11yescharacteristic
172Dic127—21no—
173Dic127—21no—
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—81no—
178C2×C127#54611yescharacteristic
179C2×C127#57511yescharacteristic
180C2×C127#57711yescharacteristic
181D127—41no—
182D127—41no—
183D127—42no—
184#3307—1271nomaximal, Sylow
185C127#330711yescharacteristic, complemented, Sylow
186#553—1271no—
187#547—1271no—
188#491—1271no—
189#485—1271no—
190#483—1271no—
191#478—1271no—
192#468—1271no—
193#465—1271no—
194#461D12711yescharacteristic, complemented
195#449—1271no—
196#449—1271no—
197#442—1271no—
198#435—1271no—
199#417—1271no—
200#416—1271no—
201C2×#41—1271no—
202C2×#41—1271no—
203C2×SD16C2×D12711yescharacteristic
204C2×SD16C2×D12711yescharacteristic
205C2×D8—1271no—
206C2×M16—1271no—
207C22×C8—1271no—
208C4⋊Q8C2×D12711yescharacteristic
209#126—2541no—
210#125—1271no—
211#123—2541no—
212#123—2541no—
213#123—2541no—
214C22⋊Q8—1271no—
215C22⋊Q8—1271no—
216#121—1271no—
217#121—1271no—
218#121—2541no—
219#121—2541no—
220C4×D4—1271no—
221C4×D4—1271no—
222C4×D4—2541no—
223C4×D4C2×D12711yescharacteristic
224#117—1271no—
225C2×C4⋊C4—1271no—
226C2×C4⋊C4—2541no—
227#107—1271no—
228#106—1271no—
229#106—1271no—
230C4⋊C8C2×D12711yescharacteristic
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103C2×D12711yescharacteristic
236#103C2×D12711yescharacteristic
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241C22⋊C8—1271no—
242C22⋊C8—1271no—
243#97—1271no—
244#41—5081no—
245#41—5081no—
246#41—2542no—
247#41—2542no—
248C2×Q8C22×D12711yescharacteristic
249C2×Q8C22×D12711yescharacteristic
250C2×D4—1271no—
251C2×D4—1271no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4C22×D12711yescharacteristic
257C22×C4—21no—
258C22×C4—1271no—
259C22×C4—1271no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4—2541no—
264SD16—41no—
265SD16—41no—
266D8—2542no—
267M16—5081no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—5081no—
271C2×C8C22×D12711yescharacteristic
272C2×C8C22×D12711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—1271no—
276C4⋊C4—1271no—
277C4⋊C4—1271no—
278C4⋊C4—1271no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4—2541no—
283C4⋊C4—2541no—
284C4⋊C4C22×D12711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—1271no—
287C22⋊C4—1271no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—21no—
297C23—1271no—
298C23—1271no—
299C23—2541no—
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304D4—21no—
305D4—21no—
306D4—1272no—
307D4—10161no—
308D4—5081no—
309D4—5082no—
310D4—5082no—
311D4—5082no—
312D4—2541no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—1271no—
318C2×C4—1271no—
319C2×C4—10161no—
320C2×C4—5082no—
321C2×C4—5082no—
322C2×C4—5082no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4D4×D12711yescharacteristic
331C2×C4D4×D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2541no—
336C8—2541no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—5081no—
341C22—5081no—
342C22—2541no—
343C22—2541no—
344C22—2541no—
345C22—2541no—
346C22—2542no—
347C22—2541no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—5081no—
356C4—5081no—
357C4—5081no—
358C4—2541no—
359C4—2541no—
360C4#41×D12711yescharacteristic
361C4#1686125411yescharacteristic
362C2—81no—
363C2—5082no—
364C2—5081no—
365C2—5081no—
366C2#2255052311yescharacteristic
367C2#2255067511yescharacteristic
368C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
hypercenter#461D127
hypocenterC127#3307
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550576C2semidirect
2. #22550529C2semidirect
3. #22550244C2semidirect
4. #22550216C2semidirect
5. #22550205C2semidirect
6. #22550177C2semidirect
7. #22550119C2semidirect
8. #22550102C2semidirect
9. #22550020C2semidirect
10. #22550020C2semidirect
11. #22549986C2semidirect
12. #22549946C2semidirect
13. #22549876C2semidirect
14. #22549871C2semidirect
15. #461×C127C2semidirect
185. C127#3307semidirect
194. #461D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225505762
#225505292
#225502442
#225502162
#225502052
#225501772
#225501192
#225501022
#225500202
#225499862
#225499462
#225498762
#225498712
#461×C1272
#3307127